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You could try booting in to recovery and wiping cache, which won't remove your apps.
If that doesn't do it (and I don't really expect it to) then you'll probably need to wipe data/factory reset.
Or, if you're on a custom ROM you could try reflashing the ROM without wiping data.
If that doesn't work then reflash the ROM with full wipes.
Also, I swiped the following from a different forum, where he made his work by doing tricks with the SIM. Credit for this goes to Nate4488 on the DroidHive forum.
"OK so I have a theory for the people who flash this rom and then no matter what cannot get data. I had this happen to me after flashing ab13 I could connect to Wi-Fi but no data. I tried the toggle trick and still no luck. So I wiped again and reflashed........no help still no data. So what I did was with the phone on and booted up I removed the SIM card until a notice pops up saying mobile network will not be available until a valid SIM card is inserted. The only option it gives you is reboot now. So put the SIM card back in and hit reboot now. When it boots back up wait a minute or two and it should get data.if not try the toggle trick between GSM and LTE. Let me know if this works for anyone else. I've had data ever since. Glad I tried this before doing something like fastbooting lol"